Molecular robots works in meso-scale (from nm to sub-micro meter). In the meso-scale, characters of each molecule and cooperative movement of molecules could not be ignored. Without calculation of potentials or molecular dynamics simulations, it is difficult to predict the behaviors of the molecules in this scale. Thus, we calculated potentials to determine the GATE size, and simulated molecular dynamics of DNAs to strength reliability of our design.

+

Molecular robots works in meso-scale (from nm to sub-micro meter). In the meso-scale, characters of each molecule and cooperative movement of molecules could not be ignored. Without calculation of potentials or molecular dynamics simulations, it is difficult to predict the behaviors of the molecules in this scale. Thus, we calculated potentials to confirm the GATE effect, and simulated molecular dynamics of DNAs to strength reliability of our design.

−

<h1>Potential Calculation</h1>

+

<h1>Numerical Analysis of Electrostatic Potential</h1>

−

The phosphate groups in the backbone of DNA are negatively-charged. Because GATE is made of DNA, we can not ignore the influence of the Coulomb force. So we calculate the electric potential inside and outside the GATE.

+

Electric potential of the GATE was calculated inside and outside the GATE. {{-}}The phosphate groups in the backbone of DNA are negatively-charged. Because the GATE is made by DNA origami, the GATE has the Coulomb force.

+

<h2>Model of DNA</h2>

+

A point-charge model is applied to DNA model. Each phosphate of a nucleotide was converted into one electric beads, which has elementary electoric charge (1.602176565 x 10^−19 [C]). In DNA helix, negative charges appear along the axis of the double helix every 10.4 base pairs. {{-}}Debye–Hückel equation was used to calculate electric potential. Electric potential of each point around the GATE was deduced from summation of all the Electric Potential for every phosphate presented on DNA.{{-}}

A point-charge model is applied to DNA model. We assume the phosphate groups have negative charge,and negative charge circles the axis of the double helix once every 10.4 base pairs. we use following fomulas to calculate electric potential.

Debye–Hückel equation:

Debye–Hückel equation:

Line 23:

Line 43:

[[Image: Potential_fomula.png |left|300px]]

[[Image: Potential_fomula.png |left|300px]]

{{-}}

{{-}}

−

Debye length

+

reference: Debye length

{{-}}

{{-}}

[[Image: Debyelen.png |left|300px]]

[[Image: Debyelen.png |left|300px]]

{{-}}

{{-}}

+

+

Following GIF animation shows the scheme of a point-charge model and potential calculations.{{-}}

+

(It takes about 7 seconds to watch one cycle of the GIF animation){{-}}

+

{{-}}

+

{{-}}

+

[[Image: Helix.gif |left|500px|thumb]]

+

{{-}}

<h2>Results</h2>

<h2>Results</h2>

−

Summing up all the Electric Potential for every DNA phosphate presented on the DNA origami GATE. (used C language to output the numbers)

+

Electric potential was calculated by using C language.{{-}}

+

Under our conditions, the target DNA has 25 electric beads.{{-}}

+

All calculation were performed under these condition: Temperature 298[K], Na+ 50mM.{{-}}

+

+

<h3>Electric potential from the top of view of the GATE</h3>

+

The following figure shows electric potentials between the target DNA and the GATE from the top of view of the GATE.{{-}}

<h3>Hybridization energy of Porters overcomes electric potential of the GATE</h3>

+

+

We expected the hybridization energy of the Porters overcomes the inhibition of entrance of the GATE by electric repulsion.{{-}}

+

The Gibbs energies of hybridization was calculated by neighbor-joining methods.{{-}}{{-}}

+

The result of Gibbs energy of the fixed Porter were defined as the maximum length of each form of Porter. For examples, Porter 1 has three forms: Stretched form, 1 loop form, and 2 loop form. In this Porter 1 case, we defined Porter 1 has three Gibbs energy according to the hybridization state. Since the maximum lengths of each form were different, the minimum energy of possible length was selected.{{-}} Gibbs energies were calculated at the center of hole in the GATE.{{-}}

+

The following figure shows the calculated energy of Porters and Toehold DNA used as controls in our project.

+

+

{{-}}

+

[[Image: Porter1.png |center|500px| Three different form ]]

+

{{-}}

−

Electric potential at the z-axis(x=0, y=0).

+

{{-}}

−

[[Image: 1014x0y0potential.png |right|300px]]

+

[[Image: Gibbs kai.png |center|800px| Gibbs energies by the Porters in the GATE ]]

−

the length of the gate is 88bp, 30nm. Target base pair 25 を点電荷と仮定する

+

{{-}}

−

もっときれいなグラフに出力できないか

+

{{-}}

+

+

The Gibbs energies by the Porters shown above were summed with electric potential of the GATE.{{-}}

+

The summation of energies were calculated at the center of hole in the GATE.{{-}}

+

The result was shown in the next figure.

+

{{-}}

+

{{-}}

+

[[Image: Gibbs2.png |center|800px| Electric potential from the top of view of the GATE ]]

{{-}}

{{-}}

+

The results of calculation indicate that the outside Porter can catch the target DNA, and inner Porter can pull in the GATE step by step. On the other hand, short DNA like the toehold DNA we called n this study cannot catch the target DNA. {{-}}

<h1>MD Simulation</h1>

<h1>MD Simulation</h1>

−

We carried out molecular dynamics simulation to examine the capturing mechanism and

+

To further strengthen feasibility of our design, we examined the capturing mechanism and

For simplicity, a course-grained DNA model is used in our simulation. One DNA nucleotide is represented by one bead in the model and each bead can be hybridized with a complementary bead.

+

For simplicity, a course-grained DNA model was used in our simulation. One DNA nucleotide was represented by one bead in the model and each bead can be hybridized with a complementary bead.

The potential energy of the system includes 5 distinct contributions.

The potential energy of the system includes 5 distinct contributions.

Line 56:

Line 138:

{{-}}

{{-}}

−

The first three terms are intramolecular interactions, bonds, bond angles, and dihedral angles. In order to express the “tether like structure”, only bond interactions are considered in our DNA model.

+

The first three terms are intramolecular interactions, bonds, bond angles, and dihedral angles. These terms are important for maintaining structure. In order to express the “tether like structure”, only bond interactions are considered in our DNA model.

−

And the latter two terms are non-bonded interactions. Coulomb interactions are taken into account using the Debye-Huckel approximation which enables to internalize counterions contribution.

+

And the latter two terms are non-bonded interactions. Coulomb interactions are taken into account using the Debye-Huckel approximation which enables to internalize counter-ions contribution.

Parameters of these potentials were fit to the reference literature ; Thomas A. Knotts ''et al''. A coarse grain model of DNA .

Parameters of these potentials were fit to the reference literature ; Thomas A. Knotts ''et al''. A coarse grain model of DNA .

Line 74:

Line 156:

−

<h2> Toehold displacement of dsDNA</h2>

+

<h2> Verification of the Coarse-grained model: Toehold displacement of dsDNA</h2>

[[Image: Format_toea_toeb.jpg |right|400px]]

[[Image: Format_toea_toeb.jpg |right|400px]]

In order to test the model, here we carried out a simulation of Toehold displacement between two strands.

In order to test the model, here we carried out a simulation of Toehold displacement between two strands.

+

Toehold strands are modeled according to designs of [http://openwetware.org/wiki/Biomod/2012/TeamSendai/Experiment#Comparison_of_Porter_and_Toehold the experiment section].

Line 107:

Line 190:

−

Movie 1 shows the trajectory of each strand from the simulation. The target strand moves from Toehold A strand to Toehold B strand which are fixed on the field. This result agrees with the energy gradient.

+

Movie 1 shows the trajectory of each strand from the simulation. The target strand moves from Toehold A strand to Toehold B strand which are fixed on the field. This result agrees with the energy gradient. The result supported our coarse-grained DNA model is feasible.

<h2> Comparison of capture ability</h2>

<h2> Comparison of capture ability</h2>

−

One of constructional features of our structure ”Cell-Gate” is the use of a novel strand displacement method.

+

One of constructional features of our structure ”CELL-GATE”is the use of a novel strand displacement method, called porter system.

−

By comparing our selector strand to toehold strand, the most popular method for strand displacement, we looked at the effectiveness of our structure in terms of it's strand capturing ability.

+

By comparing our porter strand to toehold strand, the most popular method for strand displacement, we looked at the effectiveness of our structure in terms of it's strand capturing ability.

<h3> Model and Method</h3>

<h3> Model and Method</h3>

−

According to the design of the experiment section, we modeled the selector strand and the toehold strand as shown below. <br><br>

+

According to the design of [http://openwetware.org/wiki/Biomod/2012/TeamSendai/Experiment#Comparison_of_Porter_and_Toehold the experiment section], we modeled the porter strand and the toehold strand as shown below. <br><br>

[[Image: Format_selector.jpg |left|680px]]

[[Image: Format_selector.jpg |left|680px]]

{{-}}

{{-}}

Line 123:

Line 206:

Hex-cylinder is represented as the assembly of fixed electrically-charged mass points.<br><br>

Hex-cylinder is represented as the assembly of fixed electrically-charged mass points.<br><br>

−

[[Image: Format_Hexagon.jpg |left|265px]]

+

[[Image: Format_Hexagon.jpg |left|750px]]

<html><div style="clear:both;"></div></html>

<html><div style="clear:both;"></div></html>

Line 165:

Line 248:

−

Movie2 and 3 shows the result of each simulation, selector-target and toehold-target.

+

Movie2 and 3 shows the result of each simulation, porter-target and toehold-target.

We note that this simulation was carried out under a periodic boundary condition, where the size of the simulation box is 20nm×20nm×20nm. Then, the distance between the target strand and the Hex-cylinder is maintained virtually constant.

We note that this simulation was carried out under a periodic boundary condition, where the size of the simulation box is 20nm×20nm×20nm. Then, the distance between the target strand and the Hex-cylinder is maintained virtually constant.

−

One of the advantages of the selector strand is shrinking ability. The selector strand hybridizes to the target making a loop which makes it possible to extend the strand length without changing the final structure's length.

+

One of the advantages of the porter strand is shrinking ability. The porter strand hybridizes to the target making a loop which makes it possible to extend the strand length without changing the final structure's length.

−

Results obtained from this simulation show that the selector strand can catch the target strand exists outside of the Hex-cylinder and hybridize completely while the toehold strand never hybridize to the target strand in simulation time.

+

Results obtained from this simulation show that the porter strand can catch the target strand exists outside of the Hex-cylinder and hybridize completely while the toehold strand never hybridize to the target strand in simulation time.

−

We run 5 simulations for each capturing mechanism under the same conditions and results were almost the same as we first obtained.

+

We run additional 5 simulations for each capturing mechanism under the same conditions (NOTE: random seeds were different) and very similar results were obtained. Thus, the simulation results are very feasible under our model.

−

By considering results of electrostatic potential calculation around the hex-cylinder and MD simulation, it is clear that the electrostatic field prevents the entrance of DNA strands into the Hex-cylinder and the selector strand helps it to get into the cylinder.

+

By considering results of electrostatic potential calculation around the hex-cylinder and MD simulation, it is clear that the electrostatic field prevents the entrance of DNA strands into the Hex-cylinder and the porter strand helps it to get into the cylinder.

−

Therefore, we concluded that our novel selector strand provides a high capture ability to our system “Cell-Gate”.

+

Therefore, we concluded that '''our novel porter strand provides a high capture ability to our system “CELL-GATE”.'''

Aim of Simulation

Molecular robots works in meso-scale (from nm to sub-micro meter). In the meso-scale, characters of each molecule and cooperative movement of molecules could not be ignored. Without calculation of potentials or molecular dynamics simulations, it is difficult to predict the behaviors of the molecules in this scale. Thus, we calculated potentials to confirm the GATE effect, and simulated molecular dynamics of DNAs to strength reliability of our design.

Numerical Analysis of Electrostatic Potential

Electric potential of the GATE was calculated inside and outside the GATE. The phosphate groups in the backbone of DNA are negatively-charged. Because the GATE is made by DNA origami, the GATE has the Coulomb force.

Model of DNA

A point-charge model is applied to DNA model. Each phosphate of a nucleotide was converted into one electric beads, which has elementary electoric charge (1.602176565 x 10^−19 [C]). In DNA helix, negative charges appear along the axis of the double helix every 10.4 base pairs. Debye–Hückel equation was used to calculate electric potential. Electric potential of each point around the GATE was deduced from summation of all the Electric Potential for every phosphate presented on DNA.

Debye–Hückel equation:

reference: Debye length

Following GIF animation shows the scheme of a point-charge model and potential calculations.
(It takes about 7 seconds to watch one cycle of the GIF animation)

Results

Electric potential was calculated by using C language.
Under our conditions, the target DNA has 25 electric beads.
All calculation were performed under these condition: Temperature 298[K], Na+ 50mM.

Electric potential from the top of view of the GATE

The following figure shows electric potentials between the target DNA and the GATE from the top of view of the GATE.
The electric potentials were standardized by thermal energy, KBT (KB; Boltzmann constant).
Potential energies were shown by a heat map.

Potential energy along axis indicated by the arrows are shown as the following figure.
Together, electric potential is too high for the target DNA to enter inside the GATE.

Electric potential at the center of hole in the GATE

The difficulty to go through inside hole of the GATE was also indicated by following figures.
The next figures shows electric potential at the center of hole in the GATE.
Potentials were calculated along the axis indicated by the red arrow.
Blue lines shows electric potential in the case of 1.5 fold radius of the GATE.

By the figure, we concluded
i ) Our designed size is suitable size for the GATE function
ii) Enlarging radius of the GATE decreases electric repulsion effect

Hybridization energy of Porters overcomes electric potential of the GATE

We expected the hybridization energy of the Porters overcomes the inhibition of entrance of the GATE by electric repulsion.
The Gibbs energies of hybridization was calculated by neighbor-joining methods.

The result of Gibbs energy of the fixed Porter were defined as the maximum length of each form of Porter. For examples, Porter 1 has three forms: Stretched form, 1 loop form, and 2 loop form. In this Porter 1 case, we defined Porter 1 has three Gibbs energy according to the hybridization state. Since the maximum lengths of each form were different, the minimum energy of possible length was selected. Gibbs energies were calculated at the center of hole in the GATE.
The following figure shows the calculated energy of Porters and Toehold DNA used as controls in our project.

The Gibbs energies by the Porters shown above were summed with electric potential of the GATE.
The summation of energies were calculated at the center of hole in the GATE.
The result was shown in the next figure.

The results of calculation indicate that the outside Porter can catch the target DNA, and inner Porter can pull in the GATE step by step. On the other hand, short DNA like the toehold DNA we called n this study cannot catch the target DNA.

MD Simulation

To further strengthen feasibility of our design, we examined the capturing mechanism and
the effectiveness of our structure “CELL GATE" by molecular dynamics (MD) simulation.

DNA Model

For simplicity, a course-grained DNA model was used in our simulation. One DNA nucleotide was represented by one bead in the model and each bead can be hybridized with a complementary bead.

The potential energy of the system includes 5 distinct contributions.

The first three terms are intramolecular interactions, bonds, bond angles, and dihedral angles. These terms are important for maintaining structure. In order to express the “tether like structure”, only bond interactions are considered in our DNA model.

And the latter two terms are non-bonded interactions. Coulomb interactions are taken into account using the Debye-Huckel approximation which enables to internalize counter-ions contribution.

Parameters of these potentials were fit to the reference literature ; Thomas A. Knotts et al. A coarse grain model of DNA .

The force on bead i is given by a Langevin equation

The first term donates a conservative force derived from the potential Vtot and the second is a viscosity dependent friction.

The third term is a white Gaussian noise and effects of collision with solvent molecules which causes brownian motion are internalized in this term.

Langevin equation is integrated using a Velocity-Verlet method.

Verification of the Coarse-grained model: Toehold displacement of dsDNA

In order to test the model, here we carried out a simulation of Toehold displacement between two strands.

Results

Movie 1 shows the trajectory of each strand from the simulation. The target strand moves from Toehold A strand to Toehold B strand which are fixed on the field. This result agrees with the energy gradient. The result supported our coarse-grained DNA model is feasible.

Comparison of capture ability

One of constructional features of our structure ”CELL-GATE”is the use of a novel strand displacement method, called porter system.

By comparing our porter strand to toehold strand, the most popular method for strand displacement, we looked at the effectiveness of our structure in terms of it's strand capturing ability.

Model and Method

According to the design of the experiment section, we modeled the porter strand and the toehold strand as shown below.

<html>

</html>

Hex-cylinder is represented as the assembly of fixed electrically-charged mass points.

Movie2 and 3 shows the result of each simulation, porter-target and toehold-target.

We note that this simulation was carried out under a periodic boundary condition, where the size of the simulation box is 20nm×20nm×20nm. Then, the distance between the target strand and the Hex-cylinder is maintained virtually constant.

One of the advantages of the porter strand is shrinking ability. The porter strand hybridizes to the target making a loop which makes it possible to extend the strand length without changing the final structure's length.

Results obtained from this simulation show that the porter strand can catch the target strand exists outside of the Hex-cylinder and hybridize completely while the toehold strand never hybridize to the target strand in simulation time.

We run additional 5 simulations for each capturing mechanism under the same conditions (NOTE: random seeds were different) and very similar results were obtained. Thus, the simulation results are very feasible under our model.

By considering results of electrostatic potential calculation around the hex-cylinder and MD simulation, it is clear that the electrostatic field prevents the entrance of DNA strands into the Hex-cylinder and the porter strand helps it to get into the cylinder.

Therefore, we concluded that our novel porter strand provides a high capture ability to our system “CELL-GATE”.